Commutative algebra

Yoshifumi Tsuchimoto

LEMMA 04.1   Let $A$ be a commutative ring. Then for the polynomial ring $A=A[X_1,X_2,\dots,X_n]$ of $n$ -variables over $A$ , the module $\Omega^1_{B/A}$ of $1$ -differentials of $B$ over $A$ is equal to a free module generated by $d X_1 , d X_2,\dots d X_n$ . Namely, we have

$$\Omega^1_{B/A}=A d X_1 \oplus A d X_2 \oplus\dots \oplus A d X_n.$$

LEMMA 04.2   Let $k$ be a ring. Let $A,B$ be $k$ -algebras. Then for any $k$ -algebra homomorphism $\varphi:A\to B$ we have

$$B\otimes_A \Omega^1_{A/k} \to \Omega^1_{B/k}\to \Omega^1_{B/A} \to 0$$

LEMMA 04.3   Let $A$ be a commutative ring. Let $B$ be a commutative $A$ -algebra. Then for any ideal $I$ of $B$ , we have the following exact sequence:

$$I/I^2\to \Omega^1_{B/A}/I \Omega^1_{B/A}\to \Omega^1_{(B/I)/A} \to 0$$

where the first arrow maps $f \mod I^2$ to $df$ .